Group_01_Project2_demo
Group_01
Introduction
Data come from the FIES (Family Income and Expenditure Survey) recorded in the Philippines. The survey, which is undertaken every three years, is aimed at providing data on family income and expenditure. The data obtained from this survey are from different regions across the Philippines. This report will focus on one individual area, the Cordillera Administrative Region and so region has been removed from the dataset as it will not be informative as an explanatory variable.
The report will investigate which household related variables influence the number of people living in a household. The data used consists of 1725 observations of ten variables, two of which are categorical and the remaining are numerical.
Exploratory Data Analysis
Figure 1 shows the distribution of the response variable: Number of members in a household (variable name “Total.number.of.family.members”). The modal response is 4 members and the distribution is right-skewed,
Distribution of Response Variable
The summary below shows the count data for each level of the response variable and the percentage of total households in the region in each group.
Figure 2 shows a graphical visualisation for all the variables in the data set.
Graphical Summaries of Variables
Table 1 shows summary data for all the numerical variables.There is no missing data within these variables and so no values will need to be imputed for the analysis in the report. The response variable, total number of family members in a household, ranges from 1 to 15, with the middle 50% of number of family members falling between 3 and 6 also an average number of family members of 4.67. There appear to be possible outliers at the maximum values of Total Household Income and House Floor Area. The total household income is range from 11988 to 6042860 Philippine Peso. The middle 50% of total household income is between 118565 and 328335, with an income of 269540.48 peso on average. The third variable is total food expenditure, in Phillipine peso, which is the range of 6781 to 327724, with the middle 50% lies between 51922 and 98493. Then, the household head’s age is range from 17 to 99 years, with the middle 50% falling between 41 and 63 years of age. Next, the house floor area is range from 5 to 900 square metres. The central 50% of the variable house floor area is between 32 and 102 with an average area of 90.92. The sixth explanatory variable is the house (building) age and it ranges in value from 0 to 100, with the middle 50% falling between 12 and 31. The number of bedrooms in the house ranges from 0 to 9 with a mean average number of bedrooms of 2.26 per household. Finally, we may look at the binary variable electricity, which denotes whether the property has electricity access or not, the average score of electricity is 0.93, which means 93% household have electricity.
| Variable | Missing | Complete | Mean | SD | Min. | 1st Q. | Median | 3rd Q. | Max. |
|---|---|---|---|---|---|---|---|---|---|
| Total.Number.of.Family.members | 0 | 1 | 4.67 | 2.33 | 1.00 | 3.00 | 4.00 | 6.00 | 15.00 |
| Total.Household.Income | 0 | 1 | 26.95 | 27.46 | 1.20 | 11.86 | 18.86 | 32.83 | 604.29 |
| Total.Food.Expenditure | 0 | 1 | 8.04 | 4.12 | 0.68 | 5.19 | 7.36 | 9.85 | 32.77 |
| Household.Head.Age | 0 | 1 | 52.23 | 14.52 | 17.00 | 41.00 | 52.00 | 63.00 | 99.00 |
| House.Floor.Area | 0 | 1 | 90.92 | 99.20 | 5.00 | 32.00 | 54.00 | 102.00 | 900.00 |
| House.Age | 0 | 1 | 22.98 | 15.32 | 0.00 | 12.00 | 20.00 | 31.00 | 100.00 |
| Number.of.bedrooms | 0 | 1 | 2.26 | 1.44 | 0.00 | 1.00 | 2.00 | 3.00 | 9.00 |
| Electricity | 0 | 1 | 0.93 | 0.26 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 |
The correlation coefficient between all numerical variables are shown in Table 2. There is a moderate positive correlation (0.611) between the total household income and the household food expenditure. Additionally there is a slight positive correlation between the total household income and the number of bedrooms in the household (0.441) and the number of family members and total food expenditure (0.469). The other variables are all weakly correlated.The correlation coefficient between household Head’s age, house floor area, house age and total number of family members are negative, which shows the rise of those three variables will lead to a reduction in the expected number of family members in a household.
| Total.Number.of.Family.members | Total.Household.Income | Total.Food.Expenditure | Household.Head.Age | House.Floor.Area | House.Age | Number.of.bedrooms | Electricity | |
|---|---|---|---|---|---|---|---|---|
| Total.Number.of.Family.members | 1.000 | 0.192 | 0.469 | -0.065 | -0.014 | -0.070 | 0.072 | 0.092 |
| Total.Household.Income | 0.192 | 1.000 | 0.611 | 0.063 | 0.234 | 0.025 | 0.441 | 0.149 |
| Total.Food.Expenditure | 0.469 | 0.611 | 1.000 | -0.052 | 0.124 | 0.007 | 0.356 | 0.199 |
| Household.Head.Age | -0.065 | 0.063 | -0.052 | 1.000 | 0.091 | 0.218 | 0.154 | -0.013 |
| House.Floor.Area | -0.014 | 0.234 | 0.124 | 0.091 | 1.000 | 0.074 | 0.374 | 0.107 |
| House.Age | -0.070 | 0.025 | 0.007 | 0.218 | 0.074 | 1.000 | 0.123 | 0.085 |
| Number.of.bedrooms | 0.072 | 0.441 | 0.356 | 0.154 | 0.374 | 0.123 | 1.000 | 0.214 |
| Electricity | 0.092 | 0.149 | 0.199 | -0.013 | 0.107 | 0.085 | 0.214 | 1.000 |
Table 3 shows the summaries of the two categorical variables. Single family households make up approximately two-thirds of the survey responses in this region and only 0.5% (8) of responses came from households formed of non-related individuals. Of the 1725 households, less than a quarter (21.4%) had a female head of household.
| Household.Head.Sex | n | percent |
|---|---|---|
| Female | 369 | 21.4% |
| Male | 1356 | 78.6% |
| Total | 1725 | 100.0% |
| Type.of.Household | n | percent |
|---|---|---|
| Extended Family | 569 | 33.0% |
| Single Family | 1148 | 66.6% |
| Two or More Nonrelated Persons/Members | 8 | 0.5% |
| Total | 1725 | 100.0% |
The pairs plot in Figure 3 is colour coded to illustrate any differences between the distributions of the quantitative variables when the head of household sex is included as a factor. The plots suggest the sex of the head of household may impact the number of family members in the household and the age of the head of the household.
Pair plots and correlation between numerical variables, colour coded to show the sex of the head of household.
Figure 4 shows that an extended family household or one formed by non-related individuals is more likely to have a female head, whereas a larger proportion of single family households have male heads.
Barplot of household head’s sex by type of household
Analysis of Relationships between Covariates
The relationship between independent and dependent variables
We will explore whether nine variables related to households in the dataset have an impact on the number of people living in the house (Total.Number.of.Family.members). These nine variables are Annual household income, Annual expenditure by the household on food, Head of the household’s sex, Head of the household’s age, Type of Household, Floor area of the house, Age of the building, Number of bedrooms in the house and the presence or absence of an electricity supply to the house.
Figure 5 displays scatterplots and boxplots of the response variable versus the explanatory variables.
Scatterplots and Boxplots of each predictor against the response variable
From Figure 5, it can be seen that total food expenditure (in 10000 peso), age of the head of household, sex of the head of household and age of the house seem to have a weak effect on the number of people in the household. However, we will analyze the details below by GLM.
Gender & Age
As highlighted by the pairs plot in Figure 3, there appears to be a relationship between the sex and age of the head of the household.
The minimum and maximum ages of household heads do not appear to differ greatly according to the individuals’ sex, however they do differ at the 25th, 50th and 75th percentiles with male heads of households being consitently younger than their female counterparts. The standard deviation is also greater for the female group, but the substantially smaller group size for females may contribute to this larger variation.
| Household.Head.Sex | n | Mean | St.Dev | Min | Q1 | Median | Q3 | Max |
|---|---|---|---|---|---|---|---|---|
| Female | 369 | 58.23 | 15.69 | 17 | 47 | 59 | 69 | 99 |
| Male | 1356 | 50.59 | 13.74 | 20 | 40 | 49 | 61 | 98 |
The boxplot in Figure 6 illustrates the previously summarised data. The boxplot identifies the two oldest male head of households as outliers (shown by the points above the whisker), however within the context of the data and when compared to the ages of female head of household boxplot, these ages do not appear unreasonable or unrealistic.
Boxplots of Head of Household Age stratified by Sex
The following Mann-Whitney U-test shows that there is a statistically significant difference in the median ages of male and female head of households at a 5% level.
Wilcoxon rank sum test with continuity correction
data: data.gender$Household.Head.Age by data.gender$Household.Head.Sex
W = 324284, p-value < 2.2e-16
alternative hypothesis: true location shift is not equal to 0Household Income and Food Expenditure
Figure 7 shows a boxplot of household incomes suggests a heavily skewed distribution with many outliers at the upper end of the distribution.
Household Incomes in 10000 Phillipine Pesos.
The boxplot in Figure 8 shows the log transformed household income and shows there are still several outliers following the transformation.
Boxplot of log transformed Incomes.
The scatterplot in Figure 9 of total household income against total food expenditure suggests a moderate positive correlation but the fitted model may be being heavily influenced by the extreme values, particularly by one extreme point.
Scatterplot of Income against Food Expenditure.
Figure 9 again highlights a possible outlier in terms of income, this could be a data entry error or just an outlier at the maximum. Removing this observation from the data set and plotting provides the following scatter diagram in Figure 10. This plot reconfirms the suggested positive correlation, but there is still an imbalance in the amount of data available at different levels of income. For example, most data is available for incomes between 0 and 750000 peso, but far fewer data points occur above this income level.
Scatterplot of Income and Food Expenditure with extreme value removed.
Number of Family Members & Head of Household Sex
Number of Members in Household by Sex of Head of Household
Hence we can see from Figure 11 that households with a male head appear to have a greater number of family members on average than those with a female head, as the male group has larger values for the first and third quartiles and the median. However there is overlap between the two groups central IQR and so the distributions may not be significantly different.
Figure 12 shows that an extended family household or one formed by non-related individuals is more likely to have a female head, whereas a larger proportion of single family households have male heads.
Barplot of household head’s sex by type of household
Exploratory Model Analysis
GLM Model Exploration
Prior to exploring any models, the outlier for Total Household Income and corresponding measurements for the other variables from this individual are removed.
The following code identifies which explanatory variables would be included to produce the best models of different sizes, in this instance the maximum number of variables specified is ten. The output suggests the first predictor to be included is the total food expenditure in 10000 Phillipine pesos, and the last to be included is the binary variable Electricity that identifies if a household has electricity. Comparing each of the ten models produced by BIC, CP and adjusted R^2 criteria is inconclusive as each implies a different model is best.
Subset selection object
Call: regsubsets.formula(Total.Number.of.Family.members ~ ., data = data,
nvmax = 10)
10 Variables (and intercept)
Forced in Forced out
Total.Household.Income FALSE FALSE
Total.Food.Expenditure FALSE FALSE
Household.Head.SexMale FALSE FALSE
Household.Head.Age FALSE FALSE
Type.of.HouseholdSingle Family FALSE FALSE
Type.of.HouseholdTwo or More Nonrelated Persons/Members FALSE FALSE
House.Floor.Area FALSE FALSE
House.Age FALSE FALSE
Number.of.bedrooms FALSE FALSE
Electricity FALSE FALSE
1 subsets of each size up to 10
Selection Algorithm: exhaustive
Total.Household.Income Total.Food.Expenditure Household.Head.SexMale
1 ( 1 ) " " "*" " "
2 ( 1 ) " " "*" " "
3 ( 1 ) " " "*" "*"
4 ( 1 ) "*" "*" "*"
5 ( 1 ) "*" "*" "*"
6 ( 1 ) "*" "*" "*"
7 ( 1 ) "*" "*" "*"
8 ( 1 ) "*" "*" "*"
9 ( 1 ) "*" "*" "*"
10 ( 1 ) "*" "*" "*"
Household.Head.Age Type.of.HouseholdSingle Family
1 ( 1 ) " " " "
2 ( 1 ) " " "*"
3 ( 1 ) " " "*"
4 ( 1 ) " " "*"
5 ( 1 ) " " "*"
6 ( 1 ) " " "*"
7 ( 1 ) "*" "*"
8 ( 1 ) "*" "*"
9 ( 1 ) "*" "*"
10 ( 1 ) "*" "*"
Type.of.HouseholdTwo or More Nonrelated Persons/Members
1 ( 1 ) " "
2 ( 1 ) " "
3 ( 1 ) " "
4 ( 1 ) " "
5 ( 1 ) " "
6 ( 1 ) " "
7 ( 1 ) " "
8 ( 1 ) " "
9 ( 1 ) "*"
10 ( 1 ) "*"
House.Floor.Area House.Age Number.of.bedrooms Electricity
1 ( 1 ) " " " " " " " "
2 ( 1 ) " " " " " " " "
3 ( 1 ) " " " " " " " "
4 ( 1 ) " " " " " " " "
5 ( 1 ) " " "*" " " " "
6 ( 1 ) " " "*" "*" " "
7 ( 1 ) " " "*" "*" " "
8 ( 1 ) "*" "*" "*" " "
9 ( 1 ) "*" "*" "*" " "
10 ( 1 ) "*" "*" "*" "*" Adj.R2 CP BIC
9 8 6 Formal Model Analysis
Poisson Regression model
The response variable of the Total Number of Family Members (or members of the household) can be viewed as a count and therefore a Poisson Regression model is considered. For a Poisson model to be suitable, the mean and variance should be equal and so these assumptions are checked first.
| Mean | 4.669373 |
| Variance | 5.446049 |
From Table 5, we see the variation of total number of family members is only marginally larger than the mean of total number of family members, thus, the possibility of over-dispersion in our model is not a significant issue.
Call:
glm(formula = Total.Number.of.Family.members ~ Total.Household.Income +
Total.Food.Expenditure + Household.Head.Age + House.Floor.Area +
House.Age + Number.of.bedrooms + Electricity + Household.Head.Sex +
Type.of.Household, family = poisson(link = "log"), data = data)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.7749 -0.6993 -0.1044 0.4989 3.7501
Coefficients:
Estimate Std. Error
(Intercept) 1.4254422 0.0796515
Total.Household.Income -0.0022045 0.0006612
Total.Food.Expenditure 0.0500316 0.0035528
Household.Head.Age -0.0025205 0.0008707
House.Floor.Area -0.0001932 0.0001281
House.Age -0.0023168 0.0007735
Number.of.bedrooms -0.0145830 0.0095600
Electricity 0.0276347 0.0475502
Household.Head.SexMale 0.2202770 0.0297157
Type.of.HouseholdSingle Family -0.3481835 0.0248020
Type.of.HouseholdTwo or More Nonrelated Persons/Members -0.1444455 0.1598841
z value Pr(>|z|)
(Intercept) 17.896 < 2e-16 ***
Total.Household.Income -3.334 0.000856 ***
Total.Food.Expenditure 14.082 < 2e-16 ***
Household.Head.Age -2.895 0.003793 **
House.Floor.Area -1.509 0.131385
House.Age -2.995 0.002744 **
Number.of.bedrooms -1.525 0.127155
Electricity 0.581 0.561126
Household.Head.SexMale 7.413 1.24e-13 ***
Type.of.HouseholdSingle Family -14.039 < 2e-16 ***
Type.of.HouseholdTwo or More Nonrelated Persons/Members -0.903 0.366293
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2024.3 on 1723 degrees of freedom
Residual deviance: 1330.4 on 1713 degrees of freedom
AIC: 7011.6
Number of Fisher Scoring iterations: 4The poisson model fitted with all possible covariates concludes there are six statistically significant predictors at the 5% level. These are the total household income and food expenditure, the age and gender of the head of the household, the age of the house and if it is a single family household. Table 6 shows the estimates and the lower and upper bounds of the 95% confidence intervals for the regression parameters. The rows containing significant predictors, and so where the confidence intervals do not include 0, are highlighted. We can clearly see that the confidence intervals for area, number of bedrooms, and electricity access include 0. This indicates that these three variables have essentially no effect on the number of people in the household, which exactly validates the results for the p-value as well.
| Lower Bound | Estimate | Upper Bound | |
|---|---|---|---|
| (Intercept) | 1.2687606 | 1.4254422 | 1.5810043 |
| Total.Household.Income | -0.0035110 | -0.0022045 | -0.0009192 |
| Total.Food.Expenditure | 0.0430507 | 0.0500316 | 0.0569775 |
| Household.Head.Age | -0.0042279 | -0.0025205 | -0.0008148 |
| House.Floor.Area | -0.0004469 | -0.0001932 | 0.0000552 |
| House.Age | -0.0038392 | -0.0023168 | -0.0008069 |
| Number.of.bedrooms | -0.0333633 | -0.0145830 | 0.0041111 |
| Electricity | -0.0644803 | 0.0276347 | 0.1219537 |
| Household.Head.SexMale | 0.1623556 | 0.2202770 | 0.2788479 |
| Type.of.HouseholdSingle Family | -0.3967514 | -0.3481835 | -0.2995260 |
| Type.of.HouseholdTwo or More Nonrelated Persons/Members | -0.4743426 | -0.1444455 | 0.1540719 |
We refit the model to include just the previously identified significant covariates and again evaluated the 95% confidence intervals for the estimated parameters, these values can be seen in Table 7. The intercept term of 1.436 is simply a positional constant due to the context of the variables. The negative coefficient of Total Household Income shows that for every additional 10000 peso, the number of household members is expected to decrease by 0.002. The coefficient of Total Food Expenditure suggests that for an increase of 10000 peso in spending, there is an expected 0.048 more members in the household. The coefficients of Head of Household age and the Age of the Building are both negative (-0.003 and -0.002 respectively) showing that an older head of the household or older building is linked to fewer members in a household. A Single Family household is expected to have 0.350 fewer members than the baseline category of an extended family household, and households with a male head will have 0.222 members more than their female counterparts.
Call:
glm(formula = Total.Number.of.Family.members ~ Total.Household.Income +
Total.Food.Expenditure + Household.Head.Age + House.Age +
Household.Head.Sex + Type.of.Household, family = poisson(link = "log"),
data = data)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.7325 -0.7095 -0.1012 0.5090 3.7690
Coefficients:
Estimate Std. Error
(Intercept) 1.4313832 0.0672809
Total.Household.Income -0.0028211 0.0006146
Total.Food.Expenditure 0.0503772 0.0035264
Household.Head.Age -0.0027721 0.0008627
House.Age -0.0024807 0.0007682
Household.Head.SexMale 0.2205681 0.0297155
Type.of.HouseholdSingle Family -0.3484617 0.0247448
Type.of.HouseholdTwo or More Nonrelated Persons/Members -0.1365104 0.1598583
z value Pr(>|z|)
(Intercept) 21.275 < 2e-16 ***
Total.Household.Income -4.590 4.44e-06 ***
Total.Food.Expenditure 14.286 < 2e-16 ***
Household.Head.Age -3.213 0.00131 **
House.Age -3.229 0.00124 **
Household.Head.SexMale 7.423 1.15e-13 ***
Type.of.HouseholdSingle Family -14.082 < 2e-16 ***
Type.of.HouseholdTwo or More Nonrelated Persons/Members -0.854 0.39313
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2024.3 on 1723 degrees of freedom
Residual deviance: 1336.7 on 1716 degrees of freedom
AIC: 7012
Number of Fisher Scoring iterations: 4It is important to note that the p-value for Family type of two or more unrelated individuals is greater than 0.05, thus indicating that a family type of two or more nonrelated persons is not a statistically significant explanatory variable, but households composed of a single family (p value <0.05) is a statistically significant predictor for the number of members of a household.
| Lower Bound | Estimate | Upper Bound | |
|---|---|---|---|
| (Intercept) | 1.2992842 | 1.4313832 | 1.5630233 |
| Total.Household.Income | -0.0040352 | -0.0028211 | -0.0016258 |
| Total.Food.Expenditure | 0.0434480 | 0.0503772 | 0.0572714 |
| Household.Head.Age | -0.0044639 | -0.0027721 | -0.0010820 |
| House.Age | -0.0039924 | -0.0024807 | -0.0009811 |
| Household.Head.SexMale | 0.1626469 | 0.2205681 | 0.2791383 |
| Type.of.HouseholdSingle Family | -0.3969151 | -0.3484617 | -0.2999140 |
| Type.of.HouseholdTwo or More Nonrelated Persons/Members | -0.4663624 | -0.1365104 | 0.1619512 |
The regression parameter estimates and the corresponding 95% confidence intervals are presented in Figure 13 for both the full Poisson Model and the Significant Factors Poisson Model.
Summary of Coefficients for each fitted Poisson Model
| Model | AIC | BIC |
|---|---|---|
| Full Model | 7011.64 | 7071.61 |
| Significant Factors Model | 7012.00 | 7055.62 |
From table 8, we find the AIC value of two fitted poisson models are very similar, however, the BIC value for the significant factors model is lower and so we accept the significant factors model as the better fit for the data.
The poisson regression model that will be fitted to the data is as follows: \[log(\tilde{\lambda_i}) = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \beta_3 x_{3i} + \beta_4 x_{4i} + \widehat{\beta}_{\mbox{M}} \cdot \mathbb{I}_{\mbox{M}}(i) + \epsilon_i, ~~~~ \epsilon \sim N(0, \sigma^2),\] where
- \(log(\tilde{\lambda_i})\) is the Total Number of Household Members of the \(i^{th}\) household;
- \(x_{1i}\) is the Total Household Income in 10000 Philippine pesos of the \(i^{th}\) household;
- \(x_{2i}\) is the Total Food Expenditure in 10000 Philippine pesos of the \(i^{th}\) household;
- \(x_{3i}\) is the Household Head’s Age in years of the \(i^{th}\) household;
- \(x_{4i}\) is the Building Age in years of the \(i^{th}\) household;
- \(\alpha\) is the intercept and positions the best-fitting plane in 3D space;
- \(\beta_1\) is the coefficient for the first explanatory variable \(x_1\);
- \(\beta_2\) is the coefficient for the second explanatory variable \(x_2\);
- \(\beta_3\) is the coefficient for the third explanatory variable \(x_3\);
- \(\beta_4\) is the coefficient for the fourth explanatory variable \(x_4\);
- \(\widehat{\beta}_{\mbox{M}}\) is the difference in the mean total number of household members in a household with a male head relative to a female head;
- \(\epsilon_i\) is the \(i^{th}\) random error component; and
- \(\mathbb{I}_{\mbox{F}}(i)\) is an indicator function such that
\[\mathbb{I}_{\mbox{M}}(i)=\left\{ \begin{array}{ll} 1 ~~~ \mbox{if the household head of} ~ i \mbox{th observation is Male},\\ 0 ~~~ \mbox{Otherwise}.\\ \end{array} \right.\]
Predicted Numbers of Household Members and Food Expenditure
Observed Numbers of Household Members and Food Expenditure
Predicted Numbers of Household Members against Observed Number of Household Members
Conclusions
Combining the p-values and confidence intervals for each variable, we can conclude that Total Household Income, Total Food Expenditure, Head of Household Age and Sex, Age of the House, and if the Household is a Single or Extended Family, all have an effect on the number of members in a household.